Panos Parpas

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Computational Finance

• Panos Parpas

Imperial College London
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Computational Finance Course

• Contact
• Panos Parpas (Huxley Building, Room 347)
• Email pp500_at_doc.ic.ac.uk
• and tutorial helpers.
• Look at the web for lecture notes and tutorials
  http//www.doc.ic.ac.uk/pp500
• Course material courtesy of Nalan Gulpinar.

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Course will provide

• to bring a level of confidence to students to the
  finance field
• an experience of formulating finance problems
  into computational problem
• to introduce the computational issues in
  financial problems
• an illustration of the role of optimization in
  computational finance such as single period
  mean-variance portfolio management
• an introduction to numerical techniques for
  valuation, pricing and hedging of financial
  investment instruments such as options

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Useful Information

• The course will be mainly based on lecture notes
• Recommended Books
• D. Duffie, Dynamic Asset Pricing Theory,
  Princeton University Press, 1996.
• E.J., Elton, M.J. Gruber, Modern Portfolio Theory
  and Investment Analysis, 1995.
• J. Hull, Options, Futures, and Other Derivative
  Securities, Prentice Hall, 2000.
• D.G. Luenberger, Investment Science, 1998.
• S. Pliska, Discrete Time Models in Finance, 1998.
• P. Wilmott, Derivatives The Theory and Practice
  of Financial Engineering, 1998.
• P. Wilmott, Option Pricing Mathematical Models
  and Computation, 1993.
• Two course works
• MEng test - for MEng students
• Final exam - for BEng, BSci, and MSc students

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Contents of the Course

1. Introduction to Investment Theory
2. Bonds and Valuation
3. Stocks and Valuation
4. Single-period Markowitz Model
5. The Asset Pricing Models
6. Derivatives
7. Option Pricing Models Binomial Lattices

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Introduction to Investment Theory

381 Computational Finance

• Panos Parpas

Imperial College London
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Topics Covered

• Basic terminology and investment problems
• The basic theory of interest rates
• simple interest
• compound interest
• Future Value
• Present Value
• Annuity and Perpetuity Valuation

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Terminology

• Finance commercial or government activity of
  managing money, debt, credit and investment
• Investment the current commitment of resources
  in order to achieve later benefits
• present commitment of money for the purpose of
  receiving more money later invest amount of
  money then your capital will increase
• Investor is a person or an organisation that buys
  shares or pays money into a bank in order to
  receive a profit
• Investment Science application of scientific
  tools to investments
• primarily mathematical tools modelling and
  solving financial problem
• optimisation
• statistics

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Basic Investment Problems

• Asset Pricing known payoff (may be random)
  characteristics, what is the price of an
  investment?
• what price is consistent with other securities
  that are available?
• Hedging the process of reducing financial
  risks for example an insurance you can protect
  yourself against certain possible losses.
• Portfolio Selection to determine how to compose
  optimal portfolio, where to invest the capital so
  that the profit is maximized as well as the risk
  is minimized.

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Terminology

• Cash Flows
• If expenditures and receipts are denominated in
  cash, receipts at any time period are termed cash
  flow.
• An investment is defined in terms of its
  resulting cash flow sequence
• amount of money that will flow TO and FROM an
  investor over time
• bank interest receipts or mortgage payments
• a stream is a sequence of numbers ( or ) to
  occur at known time periods
• A cash flow at
  discrete time periods t0,1,2,,n
• Example
• 1- Cash flow (-1, 1.20) means investor gets
  1.20 after 1 year if 1 is invested
• 2- Cash flow (-1500,-1000,3000)

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Interest Rates

• Interest defined as the time value of money
• in financial market, it is the price for credit
  determined by demand and supply of credit
• summarizes the returns over the different time
  periods
• useful comparing investments and scales the
  initial amount
• different markets use different measures in terms
  of year, month, week, day, hour, even seconds
• Simple interest and Compound interest

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Simple Interest

• Assume a cash flow with no risk.
• Invest and get back amount of
  after a year, at
• Ways to describe how becomes ?
• If one-period simple interest rate is
  then amount of money at the end of time
  period is
• Initial amount is called principal

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Example Simple interest

• If an investor invest 100 in a bank account that
  pays 8 interest per year, then at the end of one
  year, he will have in the account the original
  amount of 100 plus the interest of 0.08.

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Compound Interest

• Invest amount of for n years period and one
  period compound interest rate is given by
• the amount of money is computed as follows

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Simple versus Compound Interest Rates
Linear growth and Geometric growth
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Example Simple Compound Interest

• If you invest 1 in a bank account that pays 8
  interest per year, what will you have in your
  account after 5 years?
• Simple interest
• Linear growth
• Compound interest
• Geometric growth

http//www.moneychimp.com/features/simple_interest
_calculator.htm
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Example Compound Interest

• Assume that the initial amount to invest is
  A100 and the interest rate is constant. What is
  the compound interest rate and the simple
  interest rate in order to have 150 after 5 years?

Compound Interest
Simple Interest
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Compounding Continued

• At various intervals for investment of A if an
  interest rate for each of m periods is r/m, then
  after k periods
• Continuous compounding

Exponential Growth
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The effective nominal interest rate

• The effective of compounding on yearly growth is
  highlighted by stating an effective interest rate
• yearly interest rate that would produce the same
  result after 1 year without compounding
• The basic yearly rate is called nominal interest
  rate
• Example Annual rate of 8 compounded quarterly
  produces an increase

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Example Compound Interest
i ii iii iv
v Periods Interest
Ann perc. Value Effective in
year per period rate APR after 1 year
interest rate



1
6 6 1.061 1.06
6.000 2 3 6
1.032 1.0609 6.090 4
1.5 6 1.0154 1.06136
6.136 12 0.5 6
1.00512 1.06168 6.168 52
0.1154 6 1.00115452 1.06180
6.180 365 0.0164 6
1.000164365 1.06183 6.183
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Example Future Value
Suppose you get two payments 5000 today and
5000 exactly one year from now. Put these
payments into a savings account and earn interest
at a rate of 5. What is the balance in your
savings account exactly 5 years from now.
year cash inflow interest balance 0
5000.00 0.00 5,000.00 1
5000.00 250.00 10,250.00 2
0.00 512.50 10,762.50 3
0.00 538.13 11,300.63 4
0.00 565.03 11,865.66 5
0.00 593.28 12,458.94
The future value of cash flow
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Present Value (PV) - Discounting

• Investment today leads to an increased value in
  future as result of interest.
• reversed in time to calculate the value that
  should be assigned now, in the present, to money
  that is to be received at a later time.
• The value today of a pound tomorrow how much you
  have to put into your account today, so that in
  one year the balance is W at a rate of r

• 110 in a year 100 deposit in a bank at 10
  interest

• Discounting
• process of evaluating future obligations as an
  equivalent PV
• the future value must be discounted to obtain PV

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Present Value at time k
Present value of payment of W to be received k th
periods in the future
where the discount factor is
If annual interest rate r is compounded at the
end of each m equal periods per year and W will
be received at the end of k th period
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PV for Frequent Compounding

• For a cash flow stream (a0, a1,, an) if an
  interest rate for each of the m periods is r/m,
  then PV is
• PV of Continuous Compounding

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Example 1 Present Value
You have just bought a new computer for 3,000.
The payment terms are 2 years same as cash. If
you can earn 8 on your money, how much money
should you set aside today in order to make the
payment when due in two years?
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Example 2 Present Value
Consider the cash flow stream (-2,1,1,1).
Calculate the PV and FV using interest rate of
10. Example 3 Show that the relationship
between PV and FV of a cash flow holds.

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Net Present Value (NPV)

• time value of money has an application in
  investment decisions of firms
• in deciding whether or not to undertake an
  investment
• invest in any project with a positive NPV
• NPV determines exact cost or benefit of
  investment decision

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Example 1 NPV

• Buying a flat in London costs 150,000 on
  average. Experts predict that a year from now it
  will cost 175,000. You should make decision on
  whether you should buy a flat or government
  securities with 6 interest.
• You should buy a flat if PV of the expected
  175,000 payoff is greater than the investment of
  150,000
• What is the value today of 175,000 to be
  received a year from now? Is that PV greater
  than 150,000?
• Rate of return on investment in the residential
  property is

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Example 2 NPV
Assume that cash flows from the construction and
sale of an office building is as follows. Given
a 7 interest rate, create a present value
worksheet and show the net present value, NPV.
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Annuity Valuation

• Cash flow stream which is equally spaced and
  equal
• amount a1 , , an a payments per year
  t1,2,, n
• An annuity pays annually at the end of each year
• 250,000 mortgage at 9 per year which is paid
  off with a 180 month annuity of 2,535.67

Present value of n period annuity
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Annuity Valuation

• For a cash flow a1 , , an a

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Annuity Valuation
For m periods per year
The present value of growing annuity payoff
grows at a rate of g per year k th payoff is
a(1g)k
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Example Annuity
Suppose you borrow 250,000 mortgage and repay
over 15 years. The interest rate is 9 and
payments are made monthly. What is the monthly
payment which is needed to pay off the mortgage?
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Perpetuity Valuation

• perpetuities are assets that generate the same
  cash flow forever
• pay a coupon at the end of each year and never
  matures
• annuity is called a perpetuity when number of
  payments becomes infinite
• For m periods per year
• Present value of growing perpetuity at a rate of
  g